COINCIDENCE AND COSMOS

[The following is from a letter sent to a friend, who had reported a troubling coincidence, some years ago.]

~

COINCIDENCE AND COSMOS

I don’t see those two coincidences – yours or mine – as particularly startling.  But neither my being unimpressed, nor your being impressed, should weigh particularly heavily in the epistemological balance-pan.  For mankind is notoriously incapable of estimating probabilities in most instances.

One of the side benefits of faith is supposed to be  that it tends to preserve us from superstitions that might otherwise get sucked in to the vacuum where faith should be. (Chesterton was fond of emphasizing, and dramatizing, this point.)  Actually I was never superstitiously inclined, even before baptism; but now there is a warrant to just wave these things off – things superficially much more suggestive.   For, I don’t believe that God communicates via such hole-in-corner monkey-tricks.

Such incidents, when they crop up, are undeniably intriguing.  The appeal seems to be  that they hint at a pattern on the other side of the carpet, which we see only wrong-side-on.  But then, as theists, we already know that; we don’t need the occasional odd chiming of chance, to tell us so.  What is worthy only of Las Vegas, should stay in Vegas.

*

One of the things  I’ve been doing with my new-found, fiber-furnished bandwidth, is watching free online episodes of a TV series, “Lost”.   The whole thing is predicated on Baader-Meinhof phenomena.   

The problem with that  as the basis for a multi-year series, is that it is all too easy to conjure up.  Just as magic tricks are yawners if performed on television, which can always resort to special effects, so spooky coincidences are startling only if they happen to you.   It’s a very lazy genre.   For comedy to work, it has to be funny; and even a decent car-chase demands artistry to stage.  But any footling apprentice can have a stranger say (after you meet him in an empty stadium in Australia, and then part company), “See you in another life”; and then a few minutes later, a world away on a mysterious island, in a bunker far below the earth, you run into the same guy, now wild-eyed and bearded, and stammer, “Y-y-y-you….!”   Still, “Lost” not a boring show.  The whole art consists in having an artfully selected Bridge-over-San-Luis-Rey set of castaways  -- the pert freckled girl, the doughty doctor whose stubble is never shorter nor longer than a four-day growth, the Black guy, the Fat guy, the this-and-that guy – and send them through a minuet of interactions, spiced by tingly synchronicities,  so that the coincidences become tonal, as in music.

Featuring prominently among the guiding coincidences of the show is a short sequence of small integers, arranged in order.  Fat Guy overhears a mental patient (whom the numbers have driven mad) muttering them over and over, and with them, wins the lottery!  Woo-ooo!  But then very bad things start happening to everyone he goes near!! Woo-oo-ooo!  And then they turn up engraved at the entrance to that bunker!  Woo-oo-ooo-oo-oooo!  The sequence, unremarkable upon inspection…

Actually, I must confess at this point, that I am loath to write the sequence down, though it is only the whim of a TV show.  It is not superstition exactly; more like, “Get thee behind me…”  For, although God does not communicate by such monkey-tricks, the Devil might…  Anyhow, it contains an old favorite “23”.   An otherwise highly intelligent (though atheist) friend of mine  was mesmerized by this number, whose spectral footprint seemed to be everywhere.  It turns out he is not alone in his obsession; an entire movie was made (unfortunately, not a good one), about the eerie qualities of this integer.

*

There is one place where startling coincidences really are intriguing; and it is as far from Old Pagan or New Age spookery as possible.  I mean:  math and science.  For, the same underlying structures keep popping up in a variety of guises; the wild kaleidoscope of the world  appears, upon analysis, to be dreamed up out of a few symmetries  and a few bits of colored glass.

Here too it is possible to go astray, seeing significance where there is none.The great Eddington was much taken with the fact, that the Fine Structure Constant of physics (a dimensionless number, of course, otherwise its numerical value would be arbitrary) is very very close to 1/137 (or whatever the figure was).  Odd he should have noticed, this, actually; did he carry around reciprocals of all the integers in his head?  Anyhow, he hypothesized that the FSC was exactly 1/137; and busied himself attempting to explain the discrepancy as measured.  Well, it turned out to be mere gematria.  The FSC is not the reciprocal of an integer, and there’s an end to it.

The smaller the integer, the more it is likely to play a role in disparate structures essentially by happenstance.  Two is the king of them all – duality, binarism – and thus is indeed a very significant number, but rather in the way that water is a significant compound – you don’t get goosebumps when you discover another example.   Much more troubling are huge numbers, such as the ratio of the strength of the Coulomb force to that of gravity – how do you construct a cosmos out of such ill-matched yoke-mates?   Or, to take a recent example from mathematics, consider such apparently unrelated fields as the study of j-functions, and that of finite simple groups.  The first nontrivial factor in one of the series of the former is 196,884; the smallest number of dimensions in which the largest of the latter can operate, is 196,883.   A connection, or close but no cigar? 

~

Foot-note (tail-note, butt-note) anent the Dark Prince.

Two of my favorite Christian authors, G.K. Chesterton and C.S. Lewis,  offer antithetical depictions of the Devil.  Chesterton’s is more romantic and medieval:

Roses are redder  when you believe in the Devil.

Lewis’s, by contrast, in the Silent Planet trilogy, in Screwtape, and in The Great Divorce, depicts what we might call the Trivial Devil (though no less dangerous for all that).  There is no romance to him; there is, we may say, Nothing to Recommend Him.  He is no Satanic Majesty, but more like a Satanic Misery,  a Satanic Minionism -- a Mere Mechanism.  And as a mechanism, he is given to chitter-chattery repe(titi)tition.

An example of what we could term a “diabolical” coincidence, in this Lewisian sense,  occurs in “The Matrix”, when a black cat (Satan in miniature, as it might be) passes, right to left, outside the doorway, and then, right after that, or sort of seguing into it, a -- a black cat passes, right to left, outside the doorway.   Neo remarks on the coincidence, merely curious, but his more seasoned team-mates are instantly more knowing and alarmed, for they recognize a revealing glitch in the diabolical master-program that runs the Matrix.   The faults and behaviors of the dark lords who run the place, are eminently mechanical, since they are, in fact, machines.

On Crunchy Numbers

In the Ike era, we grew up on Wonderbread® :  a sort of Brot ohne Eigenschaften whose edulcorated transmogrification is known as Twinkies.

Since that time, we have learned to abjure, not only such treif, but anything not calling itself wholegrain.   Or, better still, multigrain: some brands boast seven grains, a few claim twelve; disparate mixtures  full of gritty, grainy, crunchy goodness.

 

Now  our local upscale supermarket offers a variety of own-branded bread, that boasts (in large letters) no fewer than

27 GRAINS

That really surprised me.   It’s one of the main points of Jared Diamond’s Guns, Germs, and Steel  that digestible, domesticable, feasibly growable grains  are not to be had for the asking;  there just aren’t that many of them.   The explanation is that, a little lower down and in smaller font, the label reads

AND SEEDS

In other words, this brand of bread is equally at home in the bakery and in the bird-feeder.

Our son, scoffing at this terminological legerdemain, inquired why the market chose “27” of all things.   Unhesitating I replied, “Because it is three to the third power -- the trinitarian pinnacle of the Perfect Cubes”.  -- Said heir and offspring, himself a nascent mathematician, appreciated the point, but doubted that your average shopper was aware of such things.

And indeed, there is a larger point.   For number-theorists like Ramanujan, each integer has its own flavor, its own biography and backstory -- cf. the famous incident of the taxicab numbers (in which G.H. Hardy plays the straight-man or fall-guy).  But for ordinary folks, like rocket scientists (who deal with contingent analogue quantities, rather than integral transcendent entities) or English professors (surrounded by a midge-cloud of  pre- or sub-arithmetical post-modernists), all but a very few -- small -- integers  will be featureless.   Some will be visually familiar:  3 and 4 (the triangle, the square), 5 (the quincunx on dice), 6 (boxcars, ditto),  7 (a “lucky” number for the superstitious) or 23 (ditto, for the more cerebral), 10 (count your fingers), 20 (with its portrait of Jackson).   Some will be familiar for incidental, non-mathematical reasons, like 100 and 1000 (which owe their prominence to the accident of base-10 notation -- a case of decimal fetishism).   --  Those associations are widely shared;  but there may be others  more individual;  as, (mostly for girls), “sweet sixteen” (or in Latin America, la quinceañera).   Indeed,  any integer small enough that you have lived that number of years (particularly those for which you still kept track of your birthdays).    As, the poem(-collection) “Now We Are Six” (by A.A. Milne), an anniversary in memory still green (and which I teach to the neighborhood children when they reach that delightful threshold).  Or…. 27;  which, even before I had attained that age, always seemed numinous (probably, indeed, for its prime-power nature), and which was ratified as such by my marrying at exactly that age, quite close to my birthday.    Whereas, for most folks, a number like 81 (pourtant a perfect square, as well as  3^2^2, to boot) tells no tale, sings no melody.

~

The preceding remarks are of psychological or anthropological interest, but of no moment for mathematics itself:  They present an external, human-centered view of the integers.  But further considerations suggest a subtle distinction between two ways a given integer may be “interesting” (a more bloodless equivalent of our anthropomorphic term crunchy).   We might dub these internal and external:  both times internal to mathematics as a whole, but in one case only, internal to number theory in its most elementary sense.

Thus, consider 5.  Number-theoretically, it’s a prime and that is pretty much that; but in the geometry of three-space, it is the number of Platonic solids.  Or 17:  Number-theoretically it is both a prime and a Fermat number; but in the geometry of two-space, it is the number of crystallographic planar symmetries.  

https://en.wikipedia.org/wiki/Wallpaper_group

Or: 230, the number of space groups.

Here, the integer in question is not a creature or crystal considered distinct and in itself, but a stopping-point one arrives at by calculating and counting.  There might have turned out to be, say, 18 plane symmetry groups, without upending the world; but the internal structures of 17 and 18 are unrelated.

~

This dichotomy of internal versus external interest  chez the integers, represents ideal poles, which are not exhaustive, but bookend a spectrum.  As, probably intermediate: the “crunchiness” of natural numbers as viewed by students of finite groups.    (Here the masticatory metaphor  returns unbidden, for I always imagine finite groups along the lines of a wrinkly-surfaced walnut.  Finite simple groups -- those for which no homomorphism can split out any subchunk as its “kernel” -- the hardest nuts, the ones you can’t crack.)  Simon Norton seems to have had such an intimate gustatory appreciation of individual finite groups, before he Threw It All Away and went off to ride the bus.

~

In light of our training, we know at least one thing to do, should we ever be given an unfamiliar integer and locked in a room, with no other toys to play with.  Namely, we can probe for primality.  (An activity that becomes, indeed, crucial, for cryptographers.)

But now imagine that instead we had been handed some fraction like

      (2 +  √137) /4081

Untrained, we react with dismay.

Yet later, learning of the Golden Section, (1 + √5)/2, and its many remarkable properties -- not the least of these being that it can be represented as the infinite continued fraction consisting of nothing but ones -- we conclude that the critter is crunchy indeed;  and that there are more things in heaven and earth, than are dreamt of here below, and that we must anticipate the afterlife, before we could begin to embrace them.

~

Our treatment focused on what the innumerate are missing, much like what the Daltonist, unbeknownst, lacks of the hues.   But there is such a thing as unearned crunchiness --  a bogus significance assigned to certain numinous numbers, like 19 among the Baha’i’s; the “23 enigma”; 666; 1000 AD as the Millennium; the mumbo-jumbo numbers in “Lost” and "Touch"; for all which, cf. Wikipedia on apophenia.   In the face of such things (which have snared some otherwise rational people -- a close friend of mine, critically brilliant but unmathematical, fell for the “23” business), we are inclined to say, along with Nulla  extra ecclesiam  salus,  that  Nulla  extra mathematicam  ratio.

~

For a rather recondite example of crunchiness, consider the ‘amicable numbers’  (  الآعداد المتحابة) reported by the medieval historian Ibn-Khaldun in his Muqaddima .   Apparently only two of these were known to the Arabic medievals (or: they are the only two numbers characterized by a theorem of Thabit ibn Qurra), and they are not much to look at:  220 and 284;  but they meant something to contemporary practitionars of the talismanic art.

The modern view is summarized here:

https://en.wikipedia.org/wiki/Amicable_numbers

“When I’m Sixty-Four”

I always half looked-forward to this day:  8  × 8,  a perfect square.  And of which those two factors are themselves perfect cubes.

Anyhow, in answer to the Beatles’ question:  Yes she still needs me, yes she still feeds me.   My wife and I are very much in love.

Our BFFs the Integers

(That should probably be BFsF, on the model of attorneys-general;  but let it pass.)

Richard Dedekind wrote a book with the delightful title Was Sind und Was Sollen die Zahlen? 

He proved that “the natural numbers are uniquely characterized by their induction properties.” (Wiki).

~

Elementary arithmetic -- the times table, long division -- you might find fun or you might find rebarbative, depending on the way you’re wired.  But every child loves learning to count.  It’s like the alphabet song only better;  because, unlike with the alphabet, you can keep going if you like.

Our pleasure in integers  traces back to our time in the nursery, playing with blocks.

My approach to the integers  is that of the milkmaid to the udder.  (Saying a blessing before she begins.)

Notice that the word integer is related to the word integrity.  Integers are our friends!

~

ONE little TWO little

THREE little Indians;

FOUR little FIVE little

SIX little Indians;

SEVEN little EIGHT little   NINE

little Indians:

TEN  little   In-di-an

Boys !!!

To chant that ditty

is like telling-over worry-beads --

soothed by their satisfying

click - click - click …

~

Certain integers have fan-clubs, like “23”.  That is idolatry.

One integer is much like another.   Each has various combinatorial/arithmetical properties,

but apart from primality,  it is not clear that any of these are of particular significance.

(“Perfect” numbers.  “Taxicab” numbers.)

They amuse us, is all.

They have no hidden meaning, neither individually nor collectively.

God does not speak in riddles;  gematria is false.

~

I bought some coleus the other day, each in its little pot.    I didn’t plant them right away, but set them here and tried them there, to see how they’d do.  Coleus are quite finicky about how much sunlight they’ll tolerate.  In fact, when the heat wave hit, I was glad I hadn’t planted them yet, for I brought them indoors for the cool and the shade.  But eventually I had to plant them, lest they become root-bound:  finally deciding on some spots beneath the skirts of a bushy shrub, where they are hard to see but at least won’t bake.  It took some doing.  That, and they are thirsty plants, clamoring to be watered.   Taking on coleus is a responsibility;  it’s like tending to a pet.

You can say this for the integers:  They don’t need much maintenance.

Numerology Porn

Tonight saw the debut of a new series on Fox, called “Touch”.    For this viewer, three features seemed promising:

(1)  It is in the tradition of the Paranoid Thriller, where there is a deeper significance to everything, where tout se tient -- ou se tiendrait, if only we could decipher what was going on.

(2)  It stars Jack Bauer.

(3)  It centers on Numbers.

Now, this last is in line with our theme of Our Friends the Integers, so we nursed a fragile hope, despite the precedent of the flat and mindless use of supposedly numinous numbers on “Lost” (along with the downspiral of that initially eye-pleasing series  into rank incoherence).   And despite the disappointment of last year’s Rubicon,  which shares some motifs, and showed a glimmer of promise;  but all eyes were dry when the series was canceled.

So, “Touch”… How is it?  (Or rather:  Was it;  for I shall not be watching again.)

First, I am pleased to report, the show once again exemplifies the sort of linguistic fidelity that would have been unimaginable on television during the years when I was growing up -- for them, no more than a fake/comedic ‘German’ accent, and an occasional ‘French’ ooh-la-la.  The scenes set in Iraq (which were the best, or least-worst, to my taste) were staged, not merely in Arabic, but in authentic Iraqi dialect.

For the rest, you may sift the rubble  without turning up any trinket of interest.

For one thing, there were incredibly many vastly annoying ads.  Does not Fox know that, for the Season Premiere,  you’re supposed to first sucker-in an audience, get them hooked, and only later melt their minds to the consistency of Twinkie-filling?

But the main thing:  There’s just nothing there.  Anyone can toss up a bunch of unlikely co-incidences and go Ooooooh.   As a rule, implausible coincidences count as a weakness of a plot.   To turn these into a strength  requires a very skilled hand indeed.  Thomas Pynchon did a decent job of it in The Crying of Lot 49 (though I enjoyed that book much more as an adolescent, than I did when I re-read it as an adult).   The shallow minds that put this mess together  show no such skill.

~

It is furthermore possible -- though I shall not be tracking this to see -- that the series may develop in the direction of Disability Porn.  There is a lengthy tradition of exploitative uplift, giving false hope to the suffering (whose cognitive faculties, understandably, are trumped by their emotions), e.g. “Lorenzo’s Oil.”  (I would link to Wikipedia, yet, remarkably, neither the English nor French nor German versions make any follow-up reference as to whether the stuff actually works.  Judge for yourself:

            adrenoleukodystrophy

The direction the series might take  is actually even more catagogic, suggesting that your autistic child may actually have nothing wrong with him at all, he is just “differently  abled”, in fact a geeenius, who can foil terrorist attacks  halfway around the globe  with a blink of his eyelids.

Axiomatics in its Element

[Further thoughts along the lines of the thought-themes treated in the essay that begins here.]

John Locke, An Essay Concerning Human Understanding (1690), §IV.ii.8:

The necessity of this intuitive knowledge, in each step of scientifical or demonstrative reasoning, gave occasion, I imagine, to that mistaken axiom, that all reasoning was ex praecognitis et praeconcessis;  which  how far it is mistaken, I shall have occasion to show  more at large, where I come to consider proposititions, and particularly those propositions, which are called maxims;  and to show that ‘tis by a mistake, that they are supposed to be the foundations of all our knowledge and reasonings.

Quite otherwise is the role of axiomatics in Set Theory.  Here, the axioms are once again not simply arbitrary -- they always have some prior empirical motivation.  But the consequences of assuming any new axiomatization  are initially quite obscure, and have turned out in many cases  to be enormous (or even fatal).  Hence  taking on a new axiom, or retwiddling an old one, is more akin to creating a new universe for exploration, than tidying up our formal description of the old universe.

In physics, if your axiomatization results in predictions refuted by experience -- so much the worse for that axiomatization.  Its role is not truly foundational, it’s mostly just there for show.

In logic and set theory, the case is much more complex, requiring qualitatively deeper criteria for analysis.  You wind up with logically distinct and often incompatible theories of these utterly fundamental subjects.   (Or, such we had taken them to be.  If there exists no canonical description, is their foundational character impugned?)

The proliferation of spaces (finite-dimensional and otherwise) and of surfaces or, more generally, topological objects within them, is all good clean fun -- like discovering new fauna in newly-explored islands.   But to have incompatible characterizations of logic is more like being suddenly troubled by what actually counts as a life-form.  In our own time, the biologist has had actual glimpses of such worries, in the shape of empirical discoveries not known to earlier centuries, such as viruses (alive? not alive?), and of a continuum of creaturely objects of uncertain individuation:  ant colonies; slime molds; clonal species of certain trees and fungi; and, on another dimension, organisms that divide by mitosis, or that clone themselves;  not to mention the newer view that individual genes are the fundamental vital entity, organisms being just their carrying-cases.   Now add further discoveries, such as Quantum Cats -- separate and independent twins, yet joined in an Einstein-Podolsky-Rosen way, so that they are actually not independent at all.   Or the planet in Solaris, a single organism;  and  distant cousin, the self-aware Oort Cloud.  Or:  certain subsets of integers, like the fateful one in the TV series “Lost”, with malevolent propensities of their own, and measurable biological effects.  -- The consequences for our conception of mathematical reality are quite as drastic as that.

The result is highly uncomfortable for a Platonist.  It seems as though we are coming after all  to the sterile game of creating arbitrary universes at will -- lifeless and pointless entities, with no connection to the Real (or the divine) -- a nightmare of sterile atheism.  Yet here we see one who did not flinch.  John Dawson (Logical Dilemmas, p. 175) quotes Gödel 1946 on the subject:

He asserted that, even if some new axiom ‘had no intrinsic necessity at all”, its truth might come to be accepted inductively  [dbj: Note, actual “truth”, not mere “usefulness”], on the basis of its “verifiable’ consequences” (those demonstrable without the new axiom, whose proofs by means of the new axiom are considerably simpler and easier to discover”).  Indeed, he declared, “There might exist axioms  so abundant in their verifiable consequences, shedding so much light upon the whole discipline, and furnishing such powerful methods for solving given problems… that they would have to be assumed… in the same sense as any well-established physical theory.”

The style of reasoning here strikes me as theological, quite in the tradition of those views that saw the universe, in all its parts, as having been created just so, for the well-being of God’s favorite creature, Man.  To find such an axiom as that, must indeed strike one as finding a kind of key (like that proverbial watch encountered on the empty strand, left there by the Watchmaker…)

            For:  The invention of the vacuum-cleaner certainly added spring to the housewife’s step;  but we do not therefrom conclude as to the nature of The Vacuum.    By contrast, the mathematical lawfulness of physical phenomena, in a myriad of ways, has indeed impressed observers as saying something about the Universe itself.  In similar fashion, should some axiom (with its associated methods) prove to open up, at a stroke, whole vistas of the already otherwise-intuited invisible world, we would feel it had been left there for us:  that it is as real as rocks.

~

In Modern Philosophy (1994; p.99) the philosopher and theist Roger Scruton  lists some familiar foundational principles of Truth Theory -- things like “If the sentence ‘p’ is true, then so is the sentence ‘ “p” is true’; and vice versa -- but calls these platitudes.  Very nice.  Putting axiomatics in its place.

Another way of putting them in their place  is to observe that no determinate character of ‘axiomaticity’ inheres in any one of them -- a salient illustration of Quine’s point in “Two Dogmas of Empiricism”.  Thus, from a very straightforward and standard textbook, with no philosophical (let alone relativist)  axe to grind:

What is axiom and what is theorem  is often an arbitrary choice, since one is often able to derive each logically from the other.

Sometimes one can go a step further, and make all the axioms of a mathematical system into theorems  by basing the system entirely upon some other system.  Thus, analytical geometry makes it possible to base all of plane geometry upon the properties of the real numbers.  This is also possible with R itself, throwing it back onto modern set theory.

-- Creighton Buck, Advanced Calculus (1956, 3^rd edn. 1978), p. 57

You sketch my hand,  I'll sketch yours

~

Andrew Gleason, by contrast, acutely proposes a distinction between postulates and axioms,  restoring lustre to the latter.  The distinction only makes sense  on a Realist account of mathematical truth :

When we leave the domain of abstract configurations, what we call postulates  take on a different significance.  In the abstract domain  we are in charge; we can frame postulates as we please  and simply exclude from consideration  configurations which fail to satisfy them.  But we cannot take this attitude toward concepts which have any sort of independent existence.  What are appropriately called postulates in the context of configurations  become axioms when we deal with independent conceptions.  They are axioms because they are accepted as true, or at least granted for purposes of argument, for intutitive reasons.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 153

Such a conceptual armature  in effect brings mathematics into comparison with both theology and physics, while increasing the contrast with both postmodernism and finger-painting.

Gleason is not here reporting settled usage, but putting forth a semantic proposal, one of répartition or desynonymization.  As he states earlier, anent the defining properties for an ordered set:

The conditions appearing in the definition are called the axioms or postulates for an ordered set.  The word axiom has long carried the connotation of being self-evident, but it is hard to find a sense in which these conditions are self-evident.  The word postulate (from Latin postulare, to demand) seems more appropriate.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 59

It is, incidentally, refreshing, to hear a mathematician who helped settle one of the Hilbert Problems  freely concede that a (not specially complex) set of condititions  is not, in fact, self-evident.
(In the classroom, he was very much like that:  No trace, either of arrogance or of false humility.)

~

The establishing of first principles  is not a matter for math and science only;  the widest field for its application is the law.  And as we saw in the case of mathematics and of physics, so too in law, the axioms do not historically arise first:

Primitive law is made up of simple, precise, detailed rules  for definite narrowly-defined situations.  It has no general principles.

-- Roscoe Pound, An Introduction to the Philosophy of Law (1922, 1954)

In time, such atomistic empricism is systematized.  (For a fable along these lines, consult our parable of the mathematizing woodchuck.)

In place of detailed rules, precisely determining what shall take place  upon a precisely detailed state of facts,  reliance is had upon general premises, for judicial and juristic reasoning.  These legal principles, as we call them, are made use of to supply new rules [and] to interpret old ones.

-- id.

Spinoza

Once a structured body of legal principles has arisen, judgments may be arrived at more geometrico --  though there will always be a residue of hard cases:

Judicial treatment of a controversy  is a measuring of it by a rule  in order to reach a universal solution for a class of causes, of which the cause in hand is but an example.

Administrative treatment of a situation  is a disposition of it  as a unique occurrence.

-- id.

On Gematria

 

Philo, the Jewish theologian [1^st-c. CE]  explains that God took six days to create the world because the number three stands for the male and two for the female and that  through the creative act of multiplying them  you get six.

-- Tillyard, The Elizabethan World Picture (1942)

God indeed made the integers – all the integers:  even 13, and 666.

For:

Pace superstition, 13 is simply prime, albeit rather a pricky prime.  Get to know this genus as a whole, and “13” will recede into the chorus line.

As for “666” – it is only in a purely contingent garment, the doubly-ham-handed accident of base-10 arithmetic, that this appears in such a symmetric form.  In any other base, it is an obvious hodgepodge.  To imagine that such a number possesses the least interest, is mere idolatry.  ‘Tis not the Number of the Beast – ‘tis the Number of the Moron.

[Nevertheless, for a bit of number-fun, consider this:
http://murphybros.blogspot.com/2011/10/8-8-8.html ]

And as for numerology—O ye of frigging little faith!  -- God wanna talk, he talk; no wanna talk, no talk;  but He’s not going to play some puerile hide-the-chestnut parlor-game, where He speaks in riddles, tossing out little number-puzzles  when He might have said, plainly: This;  yea, that. ….

An advantage of the Cantorian comfort with infinities  is that particular integers do not loom so large.

*

An observant Muslim of my acquaintance asked how I’d spent the weekend.  “With mathematics and religion,” I replied.

To my surprise, she brightened, and indicated her interest in numerology.

“But,” I stammered in reply, “surely such things are harâm in Islam?”  -- Not at all, she countered:  The number seven, for instance, is sacred, “because there are seven heavens, and God made the world in seven days.”

It is difficult to argue with that sort of thing.  Impossible, in fact.

She is not alone among Muslims  in indulging in that unfortunate propensity:

http://miraclesofthequran.com/mathematical_index.html

*

Well.  I won’t get into the theology of the thing; but a word on the mathematics.

Only the primes are -- primal, so to speak; whereas the composite numbers are ontologically/taxonomically one rung down, being -- literally -- the product of primes.

In this manner we built up the integers in an entirely different way, from an independent perspective.  (For one thing, here multiplication is basic;  in the successor-function approach, it’s addition.)   The result, considered as an unstructured heap, still has the same roster of individuals, but the construction is different.

Further, once you get past Kroneckerian partial-nominalism (realism about the natural numbers, nominalism about everything else), the role of integers becomes less central.  For instance, they are not at the center of point-set topology.